Will do better ∗ Fabrizio Cariani Northwestern University Paolo Santorio University of Leeds Abstract Statements about the future are central in everyday conversation and reasoning. How should we understand their meaning? The received view among philosophers treats will as a tense: in “Cynthia will pass her exam”, will shifts the reference time forward. Linguists, however, have produced substantial evidence for the view that will is a modal, on a par with must and would. The different accounts are designed to satisfy different theoretical constraints, apparently pulling in opposite directions. We show that these constraints are jointly satisfied by a novel modal account of will. On this account, will is a modal but doesn’t work as a quantifier over worlds. Rather, the meaning of will involves a selection function similar to the one used by Stalnaker in his semantics for conditionals. The resulting theory yields a plausible semantics and logic for will and vindicates our intuitive views about the attitudes that rational agents should have towards future-directed contents. [Draft of June 9, 2015] 1 Introduction Our topic is the semantics for statements about the future in English. In particular, we focus on sentences involving the English auxiliary will, such as: (1) Cynthia will pass her exam. Sentences like (1) are uniquely interesting. An account of their meaning faces challenges from a number of philosophical domains: semantics, epistemology, and metaphysics. The semantic challenge is generated by a tension in the linguistic behavior of will. On the one hand, will has the characteristic marks of a modal operator. On the other, will fails to display the standard scope interactions of modals. For example, unlike must or might, will commutes freely with negation. That is, “It will be the case that it doesn’t rain” and “It is not the case that it will rain” have the same truth conditions, despite the difference in relative scope between will and negation. ∗ For conversations and exchanges we thank [omitted for blind review] 1 The epistemological challenge comes from the role of will-statements in everyday thinking and deliberation. We are often uncertain about the propositions expressed by will-claims, and at least sometimes this uncertainty seems rational. An adequate account of will should assign will-sentences contents towards which we do, and rationally may, have attitudes of this sort. The metaphysical challenge comes from considerations about the open future. Past facts are settled, while at least some of the facts about the future seem not to be. Both the claim that the future is open and the nature of the relationship between metaphysics and semantics of the future are disputed. But the following seems uncontroversial: whatever the truth about the metaphysics, our semantic theory should avoid ascribing widespread error to ordinary speakers. The existing literature exhibits a split between philosophers and semanticists. Philosophers generally treat will as a tense, i.e. an operator whose semantic function is to shift the time of evaluation of a clause. This view is well-positioned to accommodate the epistemological challenge, and can be developed so as to meet the metaphysical challenge. But it is problematic from a linguistic point of view. Linguists (in particular, semanticists) treat will as a modal—i.e., as an expression that manipulates a world parameter. Though typical modal accounts of will are well-positioned to accommodate the linguistic challenge, they flounder in the face of the epistemological challenge. We aim to propose a new theory of will that improves on all existing accounts and draws together elements from philosophy and linguistics. Here is a sketch. Following semanticists, we hold that will is a modal. But will differs from standard modals like must or may, which work as quantifiers. The best analogy for will is the selection function meaning that Robert Stalnaker uses in his semantics of conditionals. That is, will selects a unique world among the ones that are included within a domain of quantification: roughly, this is ‘the world instantiating the one actual complete course of history’, among the ones that are compatible with history up to now. The approximate truth conditions of (1) are: (2) In the actual complete course of history, Cynthia passes her exam. Hence our semantics presupposes that there is a ‘unique’ actual course of history. At the same time, it might be indeterminate which possible world instantiates the actual course of history. As a result, it might be indeterminate which world will selects, and will-statements may have indeterminate truth value. The resulting view (i) yields a plausible semantics and logic for will, (ii) generates contents for will-statements towards which we can be rationally uncertain, and (iii) makes room for (though doesn’t require) the metaphysical claim that the future is open. 2 Will do better We proceed as follows. In §2, we outline some plausible constraints for a semantics for will. In §3, we give an informal overview of the account, which is stated in full in §5 (§4 spells out our metaphysical assumptions). In §6 we explain how to define a notion of truth that makes room for indeterminacy. Finally, we check that our account yields the desired logical and epistemological predictions (§7-8). 2 Semantics for the future: three constraints Any plausible account of will, we maintain, ought to respect three constraints. 2.1 The modal character of will Our first constraint is that will has a modal meaning. By this we mean that will manipulates a possible world parameter,1 similarly to modal auxiliaries like must or might. For example, here is a toy modal meaning for will: ðwill Añ is true at w and t iff for all worlds w0 in a relevant domain in w, A is true at w0 and at some t 0 ≥ t The modal view contrasts with a temporal view, on which will manipulates exclusively a time parameter and no world parameter. Again, for illustration, here is a toy temporal view: ðwill Añ is true at w and t iff for some t 0 ≥ t, A is true at t 0 and w To be clear: we understand the modal view as compatible with the claim that, in addition to the world parameter, will manipulates a time parameter. What distinguishes modal from nonmodal analyses is whether will manipulates a world parameter at all. The linguistics literature has provided three pieces of evidence for the modal view. Taken together, they seem to us decisive. The first piece of evidence is morphological. On one widely accepted account (Abusch 1997, 1998, Condoravdi 2002, Kaufmann 2005), will shares morphology with the modal would. In particular, will and would have in common a modal morpheme, often represented as ‘WOLL’: will is PRESENT + WOLL; would is PAST + WOLL. The assumption of common morphology allows us to explain otherwise puzzling semantic facts. For example, it explains why we can replace will with 1 It doesn’t matter to our account whether this world parameter is assumed to be represented in the object-language via world variables, or in the metalanguage via an index coordinate. When stating our semantics, we choose the latter option. 3 would in indirect reports of past utterances of will-sentences. If, on Tuesday, Harriet says “I will come to work tomorrow”, then on Wednesday we would report Harriet’s utterance by saying “Harriet said she would come to work today”. The second piece of evidence for the modal view of is that will (on a par with other expressions that normally induce future reference, like going to) may have epistemic readings (Palmer 1987, Enç 1996). These readings generally require a stative predicate (like be) in the prejacent.2 Here is an example: (3) John will be in London by now. Notice two facts about (3). First, as is made clear by the modifier by now, the prejacent of (3) has its reference time in the present. Hence, at the time of utterance, it is settled whether John is in London. Second, in (3), will works as a marker of evidentiality: roughly, it signals that the speaker is inferring John’s location on the basis of a body of evidence. To see this, notice that (3) is infelicitous if uttered by someone who is looking directly at John, even if both are indeed in London. Both these facts are hard to explain on a purely temporal view. By contrast, views according to which our language of uncertainty and prediction are both modal in nature seem ideally placed to account for them. Third, as Peter Klecha (2013) has recently argued, will allows for modal subordination. Roughly, modal subordination is the phenomenon whereby, in discourses containing several modals, earlier modals may be used to restrict the domain of later modals (Roberts 1989). As an example, consider the following discourse: (4) Jane might come to the party. Sally would come too. The occurrence of would in the second sentence is naturally understood as restricted to worlds where Jane does come to the party. A natural explanation for this is that its domain of quantification is somehow anaphoric to the worlds that witness the might-claim in (4). Now, Klecha points out that, similarly to what happens with would and other modals, will-sentences can inherit restrictions from previous modal elements of the discourse. (5) If the supplies arrive tomorrow, it will be late in the day. They will contain three boxes of cereal. This makes these sentences pattern together with modals like might and must, and differently from tenses, like the past tense: 2 Following common (and medieval) usage, we use the term prejacent to denote the clause that will takes scope over. 4 Will do better (6) 2.2 a. If the supplies arrive tomorrow, it might (/must) be late in the day. They might (/must) contain three boxes of cereal. b. #If the supplies arrived yesterday, it was late in the day. They contained three boxes of cereals. Scopelessness Our second constraint is that will is scopeless with respect to an important class of other linguistic items. By this we mean that changes in the relative syntactic scope between will and these other items don’t make a difference to the truthconditions of will-sentences. This is a remarkable feature of will, and one that is not generally shared by modal expressions. For present purposes, it is sufficient to observe scopelessness with respect to negative items.3 Consider the following pair: (7) a. b. It will not rain. It is not the case that it will rain. (7-a) and (7-b) are truth-conditionally equivalent. The situation is similar with different prejacents, and when clauses like (7-a) and (7-b) appear embedded in other environments. In short, will appears to commute freely with ordinary English negation. This observation is strengthened by considering items that lexicalize negation, such as doubt (which, following common assumptions, we understand as believe that not) and fail: (8-a-b) are truth-conditionally equivalent. (8) a. b. I doubt that Sam will pass his logic exam. I believe that Sam will fail his logic exam. For a comparison with an auxiliary that is not scopeless, consider minimal variants of (8-a-b) which involve a deontic modal. Suppose we’re talking about the obligations that Sam must fulfill in order to stay enrolled in his degree. It is clear that (9-a) is not truth-conditionally equivalent to (9-b). (9) a. b. I doubt that [in order to graduate] Sam must pass his logic exam. I believe that [in order to graduate] Sam must fail his logic exam. The lack of scope interactions with negation immediately yields an interesting logical constraint: 3 Perhaps the scopelessness of will extends further. One further category of items with respect to which will might be scopeless is comparative expressions (see the discussion of comparatives and conditionals in Korzukhin 2014). For reasons of space, we limit ourselves to considering negation. 5 Will Excluded Middle (preliminary take): ðwill A ∨ will not Añ is a logical truth For now, we informally gloss ‘logically true’ as ‘true whenever uttered’. In §7, we derive the validity of this schema, given our semantics and two standard formal concepts of consequence. 2.3 The cognitive role of future statements Future-directed statements play an important role in our cognitive economy. It is a platitude that ordinary agents are uncertain about the future. Assuming that credences attach to propositions, it seems natural to understand ‘being uncertain about the future’ as ‘having a non-extreme degrees of belief towards the propositions that are expressed by will-claims’. Moreover, at least in some cases, these non-extreme degrees of belief seem also rational. Sports Fan: Suppose that Cynthia comes to work each day wearing a Warriors cap, a Giants cap or no cap, depending on a random draw (with each option having equal probability). What degree of belief should you assign to the proposition that tomorrow she’ll wear a Warriors cap? Presumably "1/3" is a rationally permissible answer. In some theoretical settings, it may even be required: if some version of Lewis’s (1986c) principal principle is a requirement of rationality and all of your evidence is of the admissible variety, "1/3" would appear to be the only rational answer. Similarly, it seems that the fair odds for a bet on the truth of that proposition are 1 to 2. These seem like truisms, yet they are surprisingly hard to vindicate on existing modal accounts. For illustration, consider the toy modal semantics mentioned in §2.1. On that semantics, will is a universal quantifier over a domain of worlds. On a natural construal of the domain of quantification of will, on this semantics ðwill Añ says that A is true at all futures that are ‘open’ from the point of utterance (this semantic account captures what Prior (1967) calls the ‘Peircean future tense’). The problem for this semantics is that if Warriors-cap futures and Giants-cap futures are both possible, you should have zero credence in the propositions expressed by each of (10) and (11). (10) Cynthia will wear a Warriors cap tomorrow. (11) Cynthia will wear a Giants cap tomorrow. 6 Will do better Any theory that treats will as a universal quantifier faces this problem.4 2.4 Taking stock It is difficult to satisfy all three constraints. For one thing, the first two seem to be in direct tension with each other. If will is a modal, as the first constraint requires, we expect it to have nontrivial scope interactions, in violation of the second constraint. And indeed, basic modal analyses of will predict that switching the relative scope of will and negation does have truth conditional effects. For illustration: on Kaufmann’s (2005) account, will is a universal quantifier over (roughly) the worlds realizing the most likely courses of future history. On this theory, by switching around will and not, we get the two nonequivalent readings: will > not: all most likely futures do not satisfy the prejacent. not > will: not all the most likely futures satisfy the prejacent. The linguistics literature offers attempts to reconcile the first two desiderata. A prominent example is the modal analysis of Copley (2009). Building on work by von Fintel on generics (1997), Copley claims that will-sentences presuppose that their domain is homogeneous with respect to the prejacent. For an occurrence of will in a sentence of the form ðwill Añ to have a denotation, its domain must contain only A-worlds or only ¬A-worlds. We have concerns about the stipulative character of this proposal. But we can set them aside, because the theory runs into a more basic problem: it fails to address our third constraint. The propositions that this theory delivers are not propositions that we can plausibly have nonextreme credences in. To see this, consider again (1): (1) Cynthia will pass her exam. Simplifying, the basic truth conditions of (1) on both Kaufmann’s and Copley’s views are that it is settled that Cynthia will pass the exam. Now, suppose that you believe that the future is genuinely open with regard to Cynthia passing the exam: it is unsettled whether she will. Then your credence in (1) must be zero: you are certain that the truth condition of (1) fails to hold.5 4 The problem is analogous to one that has recently received attention in the counterfactuals literature (see Edgington 2008, Moss 2014, Schulz 2014). 5 We do not claim that it is impossible for Copley to extend her view to account for these facts about credences. It might well be possible to replicate in her framework some of the moves we make in §8. We just claim here that Copley’s framework cannot be taken as is to account for the cognitive role of future statement. Moreover, the solution to the cognitive problem we will devise is more straightforwardly applied to the semantics we go on to construct. 7 Let us now peek quickly at the philosophical literature. By far the most popular view among philosophers is what Prior calls ‘Ockhamist semantics’. Ockhamists don’t ascribe any modal character to will. For them, “It will rain” is true if and only if there is a future moment at which it rains (context might further narrow down the interval during which this moment is situated). The obvious problem with Ockhamism is its inability to satisfy our first desideratum. The Ockhamist has no story about the relationship between will and would, about predictive uses of will, or about modal subordination.6 One final option is to claim that will is ambiguous between a modal and a nonmodal meaning. The modal meaning explains why will seems to satisfy the first desideratum. The nonmodal meaning explains why it seems to satisfy the second and third desiderata. We won’t attempt a full refutation of the ambiguity option; but we notice that it has two major disadvantages. First, it systematically overgenerates. For example, it predicts a true and a false reading for: (12) The probability that Cynthia will wear a Warriors cap is 0. (12) is true on the modal meaning and false on the nonmodal meaning, so we should be able to hear it as true. Perhaps there are maneuvers to be made to block this reading, but we leave it to the ambiguity theorist to explain what they are. Second, an account of will that does not exploit ambiguity seems obviously preferable on the usual grounds of simplicity and theoretical unity. So, by giving an account of will that satisfies all desiderata we provide an indirect argument against the ambiguity view. 3 Overview: selecting the future We present our full account of will in the next few sections, but it is helpful to illustrate the central ideas without the formalism. We start by adopting some (but not all) of the insights associated with branching time frameworks. At every moment in time, we suppose, there are multiple possible histories that fully coincide with respect to the past and diverge with respect to the future. As time 6 It might appear that classical supervaluationism is an exception to this pattern. Classical supervaluationists have a two-step approach: first, define truth relative to a single possible future in the same way as the Ockhamist semantics. Next, say that A is true simpliciter just in case it is true at the time of utterance in every objectively possible future. However, as Cariani (unpublished) points out, supervaluationism is not a modal theory of will. Though it has a modal element, it is not distinctive of the lexical entry for will, but in the ‘global’ definition of truth at a context. For this reason, classical supervaluationism is unable to account for the evidence for a modal treatment of will we summarized in §2.1. This does not entail that supervaluationist technology is not useful. Indeed, we will avail ourselves to a non-classical form of supervaluationism in §6. 8 Will do better passes, histories that had previously coincided up to a point part ways, ‘making true’ different courses of events. In diagram form: t0 t1 t2 t3 t4 This picture is often combined with substantial metaphysical claims about the nature of possible worlds and the indeterminacy of the future. Importantly, our account is neutral between all the relevant options (we clarify this in §4). All we need is that, given any world w and time t, we can determine the historical alternatives to w at t. Here is how we do it: Two worlds w and v are historical alternatives at t iff w and v match perfectly in their history (i.e., iff they match perfectly in matters of particular fact) up to t. Note that our definition of historical alternatives involves no reference to a notion of openness. This is key to our later vindication (§4) of the claim that our semantics is neutral about the open future hypothesis. Now, consider again (10), repeated here: (10) Cynthia will wear a Warriors cap tomorrow As a first step, we assume that will is a modal. Like all modals in natural language, it is interpreted against a background set of possibilities. Following Kratzer’s terminology (1977, 1981a, 1991b), we call this set the modal base. For the particular case of will, the modal base in a given context is the set of historical alternatives to the world of the context, at the time of the context. For example, in the scenario we described, the modal base of (10) includes worlds where Cynthia wears a Warriors cap, worlds where she wears a Giants cap, and worlds where she wears no cap. In diagram form: 9 t0 Warriors cap Giants cap w t v no cap z Figure 1 Standardly, the modal base is the domain of quantification of the modal. But, as we anticipated, our account is not quantificational. Instead, we propose that will singles out one world within the modal base, and evaluates the prejacent at that world. Intuitively, the selected world represents the ‘way things will actually be’—in other words, the historical alternative that will actually be realized. So, (10) is true just in case, in the selected world, Cynthia wears a Warriors cap tomorrow. t0 t Warriors cap Giants cap w v no cap z Figure 2 The explanation for the scopelessness of will (and consequently the validation of will-excluded middle) flows immediately from the fact that the prejacent is always evaluated relative to a single world (see §7). Our semantics for will presupposes that, at the time of utterance, there is a unique, fully specified way things will actually be (in the jargon introduced by Belnap & Green 1994, this is the assumption that there is a ‘thin red line’ that marks the complete course of actual history). This assumption is controversial. Theorists in the branching time tradition object that, in the context of futuredirected discourse, we have no right to speak of ‘the way things will actually be’, or of ‘the actual world’. On the one hand, it might be that the future is open—that there is no fact of the matter, at the current time, about what way things will turn out. On the other, even if the future is not open, it is not clear that a semantics for natural language can legitimately presuppose a metaphysical claim of this sort. 10 Will do better Even if one agrees with these concerns, we don’t think that the compositional semantics for will needs to be changed. We distinguish what information is needed by the compositional semantics from what information the world is able to supply. We assume that the compositional semantics requires as input a unique world of utterance. Like all parameters used in semantic computations, the value of this parameter is supplied by the context. At the same time, we leave it open that it may be indeterminate what context the utterance takes place in, and hence what world is supplied to the semantics. For illustration, consider again (10). Perhaps, at the time of utterance, it is indeterminate whether the actual world is a Warriors-cap-world, a Giants-capworld, or a no-cap-world. If so, it is indeterminate which context the utterance of (10) takes place in. The context might be the context of Figure 2, or it might be a context in which some other world (for example, v, as in Figure 3) is selected. t0 t Giants cap Warriors cap w v no cap z Figure 3 Let us highlight an important point. We grant that it may be indeterminate what world an utterance takes place in; moreover, we said that the modal base of will is determined as a function of the world and the time of the context. But it doesn’t follow that the modal base of will is indeterminate. The reason is that, given the way that we have defined historical alternatives, all worlds that are candidates for being the world of the context have the same historical alternatives. So we will be able to speak of the modal base of will in a context, even if it is indeterminate what context the utterance takes place in. The next few sections implement the plan we just sketched. §4 specifies some metaphysical background. §5 presents our compositional semantics, including an analysis of will-conditionals. §6 elaborates on our treatment of indeterminacy; §7 shows how our account yields the logical and linguistic predictions we identified; §8 shows how our account yields appropriate predictions about cognitive role. 11 4 Metaphysical background Our account is neutral on a number of metaphysical issues connected to branching. In this brief section we explain how. First, supporters of branching time often claim that possible worlds literally share temporal initial segments (see, e.g., Thomason 1970, Belnap & Green 1994, Belnap et al. 2001). The point at which two worlds branch is the point at which the initial segment ends. By contrast, opponents of branching argue that worlds with identical histories up to a point are qualitatively identical, but still have no parts in common (see, e.g., Lewis 1986a). We understand the claim that two worlds w1 and w2 are historical alternatives at a time t as the weak claim that there is perfect match between matters of particular fact between w1 and w2 up to t. This is compatible both with genuine overlap and with mere qualitative identity. Second, the branching framework is often associated with the claim that the future is ‘open’. The relevant contrast here is with past events, which are taken to be fixed in a way in which the future is not. There are a number of ways to explain the relevant concept of openness. Following Barnes & Cameron (2009), we choose one that is noncommittal between different metaphysical theses: Openness: (at least some) contingent facts about how things will be are presently unsettled (in a suitably objective sense). Some writers (for example, Belnap & Green 1994) adopt Openness as the starting point of the enterprise of giving a semantics for will. Others (like MacFarlane 2003, 2008, 2014) take it as a methodological desideratum that a semantics for the future should not decide between different metaphysical options about Openness. We are not committed to any of these claims. Unlike Belnap and Green, we don’t assume Openness. Our apparatus is compatible with both Openness and its denial. Unlike MacFarlane, we don’t endorse the neutrality of the semantics as a methodological constraint. As it happens, however, our semantics for will does turn out to be metaphysically neutral about Openness—in the sense that both the defender and the opponent of Openness are able to use it. The reason is that, as we pointed out in §2, there is strong independent reason to adopt a modal semantics for will. This kind of modal semantics lends itself well to the sort of neutrality about Openness we just characterized. But we do not claim that the metaphysical neutrality is, in itself, a reason to accept our account. 12 Will do better 5 5.1 Semantics Setup Let us start by introducing some notation. We use sans-serif letters (‘A’, ‘B’, etc.) as metalinguistic variables over sentences; and boldface letters (‘A’, ‘B’, etc.) as metalinguistic variables over sets of worlds. We use ‘propositions’ and ‘sets of worlds’ interchangeably, but everything we say is meant to be compatible with theories according to which propositions merely determine sets of possible worlds, without being identical to them. As is standard in semantics, we define an interpretation function of the form, ¹·º par amet ers,g Such a function assigns compositional truth-values to sentences relative to a series of parameters and an assignment function, conventionally denoted by ‘g’ (the latter is just a function that assigns objects to syntactic indices, and is needed to handle variables). Different theories employ different parameters. The interpretation function is also relativized to a context, but to remove clutter we avoid explicit mention of the context unless strictly needed. We also make some specific assumptions about will. First, will is a sentential operator, i.e. an operator that takes a full clause as argument. This is a simplification, but one that is harmless given our purposes. Second, as we flagged in §3, will takes as argument a modal base, i.e. a set of worlds that are used for the interpretation of the modal.7 In particular, the modal base of will is the set of historical alternatives to the world of the context. Syntactically, we assume that modal bases are the semantic values of covert pronouns that work as arguments of modals. We represent these pronouns as ‘ f i ’, and their values, a set of worlds, as ‘Fi ’. For shorthand, we generally represent modal bases in LFs just as a subscript of modals; hence we write ‘will f ’ rather than the more extended ‘will [ f ]’. 5.2 Semantics for will Our semantics for will is based on an extended analogy with Stalnaker’s (1968) semantics for conditionals.8 7 This understanding of modal bases is slightly simplistic. Modal bases are officially functions from worlds to propositions (see von Fintel & Heim 2011 for discussion). 8 Schulz (2014) has recently defended an interesting variant of Stalnaker’s semantics. Roughly, conditionals quantify over a set of worlds, but they also select (via a choice function) an arbitrary world within that set. We lack space for a full comparison here. Let us just state without argument 13 Similarly to what happens on Stalnaker’s semantics, we assume that the interpretation of will involves appeal to a selection function, denoted by ‘s’. A selection function maps a pair of a world w and a proposition A to a ‘selected’ world w0 . Intuitively, s selects the world w0 that is ‘closest’ to the starting world w while at the same time verifying proposition A. For the case of conditionals, and counterfactuals in particular, there is much literature on how exactly the metric of closeness should be construed.9 We don’t need to settle these issues here. We can adopt any of the metrics that have been proposed for counterfactual conditionals.10 Selection functions are characterized by two important constraints: Inclusion: if A is non-empty, s(w,A) ∈ A Centering: if w ∈ A, s(w,A) = w. Inclusion says that the world selected must verify the input proposition (provided that some world does verify the input proposition). Centering says that, if the input world verifies the input proposition, then the world selected is the input world itself. Inclusion and centering are the only constraints we impose on selection functions, which can then be defined as follows:11 A function s : W × P (W ) 7→ W is a selection function iff i. if A is non-empty, s(w,A) ∈ A, and ii. if w ∈ A, then s(w,A) = w. At this point, we’re ready to state the meaning of will. We assume that interpretation is relativized to three parameters:12 a world of evaluation w, a selection function s, and an assignment g. (13) ¹will f Aºw,s,g = 1 iff ¹Aºs(w,g( f )),s,g = 1 or development that, on the most natural implementation, building the analogy with Schulz’s system rather than Stalnaker’s would require us to consider indeterminacy at the level of the compositional semantics, which we are reluctant to do. 9 For some sample proposals concerning the metric of closeness for counterfactuals, see Lewis (1979a), Kratzer (1981b), Hiddleston (2005). 10 As will be evident soon, the choice between different metrics only matters for will-conditionals. All we need to settle the selected world in all other cases is merely the Centering condition (see below). 11 Stalnaker imposes some extra constraints on selection functions. We leave it open whether these extra constraints should apply to will; nothing hinges on these for our purposes. 12 Recall: to avoid clutter, we omit the context parameter. 14 Will do better To simplify the notation, we will just write F instead of g( f ) throughout the paper. Let us make some comments about this compositional semantics. First, the basic effect of will is to shift the world at which its prejacent is evaluated. This feature is shared with standard semantic accounts of modals in natural language, like must. The difference is that modals usually introduce quantification over the world of evaluation parameter, while will replaces the world of evaluation for another one picked via the selection function. Second, the entry in (13) does not reflect any temporal shift. It is easy to introduce temporal shift, letting will quantify existentially over times (accordingly, interpretation is relativized to an extra parameter for times). (14) 0 ¹will f Aºw,t,s,g = 1 iff ∃t 0 ≥ t,¹Aºs(w,F),t ,s,g = 1 This said, throughout the discussion we just stick to the entry in (13). This is mostly for simplicity. The central innovation we introduce is the appeal to selection functions. Other elements of the meaning of will can stay in the background. Moreover, there are reasons to think that a full-blown semantics for will exploits time in a way that is more complex than simple existential quantification over temporal instants.13 So the account in (14) would need update and clarification anyway. Third, this semantics has an interesting consequence for unembedded occurrences of will: as it turns out, will is semantically vacuous with respect to the modal parameter. Recall that the modal base of will defaults to the set of historical alternatives to the world of the context. Furthermore, the initial world of evaluation defaults to the world in which the utterance takes place.14 Hence, when will is unembedded, the world that works as the input to the selection function is a member of the modal base. In this situation, the centering assumption entails that the world returned by the selection function is always the world of evaluation itself. Thus, in its unembedded occurrences, will merely ‘overwrites’ the world of evaluation parameter with itself. (15) ¹will f Aºw,s,g = 1 iff ¹Aºw,s,g = 1 13 In particular, as argued by Abusch (1998) (see also Condoravdi 2002), it seems that our best semantics for tense should quantify over intervals rather than instants. In the context of this theory, the semantic effect of will (as well as of other modals) would be not to shift the time of evaluation, but rather to extend forward the time interval at which the prejacent is evaluated (for a proposal in this vein, see also Kaufmann 2005). 14 This is via the definition of truth at a context, which fixes the value of w to the world of the context. See section 6 for details. 15 Thus, when will occurs unembedded, our semantics effectively collapses on the simple Ockhamist semantics, which treated will as a mere tense. Why bother, then, with the complexities of our selection function semantics? There are many good reasons. They mainly relate to the fact that, on our account, will has a modal base. This opens up the possibility of accounting for will-conditionals (adopting the popular assumption that will-clauses function as restrictors), modal subordination (via anaphoric links between the modal bases of the different modals), and epistemic readings (by assuming that the modal base can have different flavors). The selection function account also allows us for a vindication the will-would connection. Giving a full-fledged account of all these phenomena would take too long. But below we give a brief sketch. Even from these quick remarks, it should be clear that our account provides the tools for vindicating in full all the marks of the modal character of will. 5.3 5.3.1 Applications Conditionals and modal subordination A selection function semantics for will allows for a natural account of willconditionals. This will also allow us to account for modal subordination. Following a longstanding tradition (see e.g. Lewis 1975, Kratzer 1991a, 2012) we assume that the function of if -clauses is to restrict modal bases—to rule out of the modal base the worlds that are incompatible with them (this effect is modeled by intersecting the modal base with the set of worlds individuated by the if -clause). There are many ways to implement this semantic effect. Building on work on quantifier domain restriction by Kai von Fintel (1994), we choose a simple one that dovetails well with our assumption that the object language syntax contains a variable referring to the modal base. Nothing hinges on this particular choice of implementation. We assume that if -clauses work as assignment shifters (similarly to lambdabinders in a system in the style of Heim & Kratzer 1998). At a syntactic level, if -clauses are coindexed with the relevant modal base variable. For example, the LF of (16) is in (17): (16) If John goes to London, he will meet with Matthew. (17) [If John1 goes to London]4 will f4 [he1 meet Matthew] 16 Will do better At a semantic level, conditionals are interpreted via a rule that instructs us to perform assignment shift, mapping the modal base variable to a set of worlds determined by intersecting the old modal base with the proposition expressed by the antecedent. Formally: (18) ¹(if A)(M ODAL f B)ºw,s,g = ¹M ODAL f Bºw,s,g[ f →F∩A] (recall that F = g( f ) and A = {w : ¹Aºw,s,g = 1}) To illustrate this, consider again (16). Given modal base F, let g ∗ be the assignment that coincides with g except at index 4, which is mapped to the set of worlds in F at which John goes to London (i.e. g ∗ = g[4 → F∩ ¹John goes to Londonº]). Then we predict: (19) ¹(16)ºw,s,g = ¹will f4 [he1 meet Matthew]ºw,s,g ∗ Informally, and simplifying, the resulting truth conditions of (16) are: (20) ¹(16)ºw,s,g = true iff John meets Matthew at v where v is the world that is selected when s is given as input the set of the historical alternatives (to w) where John goes to London. Notice that the selected world need not coincide with the actual world or with the world of evaluation. In particular, for any w such that John does not go to London at w, then the world selected by s taking w as input must be different from w itself. In this case, our Stalnakerian semantics and the Ockhamist tradition diverge. This treatment of conditionals also yields a straightforward account of modal subordination. Consider again Klecha’s example: (5) If the supplies arrive tomorrow, it will be late in the day. They will contain three boxes of cereal. We can predict the relevant interpretation of (5) by assuming that the modal base variables associated to the two occurrences of will are coindexed: (21) If the supplies arrive tomorrow, it will f3 be late in the day. They will f3 contain three boxes of cereal. 17 The antecedent if the supplies arrive tomorrow shifts the value of the relevant index. But, given coindexing, both occurrences of will are interpreted in the scope of the relevant supposition.15 5.3.2 The will-would connection Our treatment of will also allows us to vindicate the morphological connection between will and would. The precise nature of this connection depends on one’s views about the meaning of would. On the one hand, if one assumes a Stalnakerian semantics for would, then the connection is immediately vindicated: will and would turn out to have exactly the same meaning—modulo differences in what possibilities are in the modal base in the two cases. Of course, Stalnaker’s semantics for would, and in particular the principle of Conditional Excluded Middle that it entails, are controversial. But, first, notice that all the arguments that we gave above for the scopelessness of will carry over to the case of would. Moreover, the literature has provided plenty of further arguments in support of Conditional Excluded Middle (see, e.g., von Fintel & Iatridou 2002, Williams 2010, Klinedinst 2011). On the other, the connection is not straightforward if one adopts a Lewisian semantics for would. In this case, one will have to explain why will deploys a selection function while would has universal quantificational force. While this is a nontrivial task, a selection function account of will is better placed at fulfilling it than an old-fashioned Ockhamist semantics. 5.3.3 Epistemic readings of will Recall the example we used to introduce epistemic readings of will: (3) John will be in London by now. will in (3) is not used to talk about the future, but rather has an epistemic reading. Predicting how and when will receives an epistemic reading goes beyond the scope of this paper. These questions connect to general phenomena in the semantics of modality, and its relationship with tense and aspect, that we can’t cover here.16 To see this, notice that the availability of epistemic readings for will patterns with the availability of epistemic readings with other modals, like must and have to. 15 We assume that the effects of the shift operated by the conditional antecedent extend beyond the boundaries of individual sentences. An assumption of this sort seems required by any account of modal subordination. 16 For some relevant discussion, see Condoravdi (2002), Condoravdi & Deo (2008). 18 Will do better (22) a. b. John will be in London by now. (Øepistemic/ #historical) John will go to London tomorrow. (#epistemic/ Øhistorical) (23) a. b. John must/has to be in London by now. (Øepistemic/ #deontic) John must/has to go to London tomorrow. (#epistemic/ Ødeontic) The natural way to account for contrasts like the ones in (22-a-b) and (23-a-b) is to assume that the different prejacents somehow force a different choice of modal bases for the modals. For example, Condoravdi (2002) suggests that modalized claims whose prejacent has a reference time in the present (like (22-a) and (23-a)) rule out non-epistemic modal bases via a constraint requiring that the modal base be sufficiently diverse.17 Even if we can’t provide an account of the data here, it should be clear that our semantics provides the tools to build such an account. Once again, the key move is endowing will with a modal base, contrary to Ockhamist semantics and in agreement with modal semantics. Once we make this move, we can just draw on independent work on the interaction between tense, aspect, and modal bases. 6 Truth, validity, and indeterminacy §5 offers a compositional semantic account of will, but does not fix the truth conditions of will-claims. To get the latter, we must define a notion of truth at a context (this is the stage of the theory that MacFarlane 2003 calls ‘postsemantics’). Adopting a definition of truth at a context requires us to take sides in the debate about the indeterminacy of statements about the future. To make the presentation more concrete, we adopt a specific account of how indeterminacy affects the semantics, i.e. the version of supervaluationism defended by Barnes & Cameron (2009) (see also Iacona 2014, who sketches an Ockhamist picture whose treatment of indeterminacy is parallel to ours). According to this form of supervaluationism, each context determines a single actual world, but it is indeterminate which context the utterance takes place in. In a standard contextualist framework, built on Kaplan (1989), truth at a context is defined by fixing the values of index parameters to the coordinates of the context (following Lewis 1980, we take contexts to be concrete situations of utterance). Here is a formal definition: Truth at a Context. A is true as uttered at c iff ¹Aºwc ,sc ,gc = 1 17 This is the constraint endorsed by Condoravdi, roughly stated: Diversity condition. If ðMODAL Añ has a non-epistemic modal base M, then there are worlds w and v in M such that A is true at w and false at v. 19 In a slogan: S is true at a context c just in case S is true at the circumstances fixed by c. Traditional supervaluationist accounts reject Kaplan’s definition of truth at a context. They maintain that a context of utterance does not fix which world is to count as actual. These accounts replace truth at a context with a new definition that allows for sentences that are neither true nor false at the context of utterance.18 Unlike traditional supervaluationists, we endorse the simple definition of truth at a context above. Hence, on our account, every sentence is either true or false at a context. How, then, can we satisfy the theorists who maintain that the future is genuinely open? We assume that if the future is open, it is indeterminate which context the utterance takes place in, and hence it is indeterminate which truth value the sentence has. An important consequence of this hypothesis is that both the defender and the opponent of Openness are able to help themselves to our framework, including the definition of truth (and the two notions of validity that we give below). Their disagreement is moved out of the semantic apparatus entirely, as the defender of Openness denies, and the opponent of Openness claims, that a concrete situation of utterance determines a unique context. The notion of truth at a context is important for a number of reasons. One of them is that (following Kaplan himself) we can use it to define a plausible notion of validity. On this notion, an argument is valid just in case it preserves truth at a context: no context makes the premises true and the conclusion false. Validity1 . A1 ,..., An 1 B iff, for any context c such that A1 ,..., An are true at c, B is also true at c. We should flag an important consequence of our moving the indeterminacy outside of the semantics: unlike traditional supervaluationism, our logic for the relevant fragment of the language is allowed to be straightforwardly classical. It will also be helpful to appeal to a second notion of validity, one that captures preservation of truth at a point of evaluation. Validity2 . A1 ,..., An 2 B iff, for any triple 〈w,s, g〉 such that ¹A1 ºw,s,g = 1, ..., ¹An ºw,s,g = 1, ¹Bºw,s,g = 1 A single sentence A is valid1 just in case 1 A; similarly for valid2 . Note that every argument that is valid2 is valid1 , but not viceversa. When A and B are such that A 1 B and B 1 A, we say that they are equivalent1 (similarly for equivalent2 ). Here too we have that every equivalent2 pair is equivalent1 (though 18 For completeness, here is a sample definition of truth at a context that fits the traditional supervaluationist’s desiderata: A is true as uttered at c iff for all worlds v that are historically possible in c , ¹Aº v,sc ,gc = 1. 20 Will do better not vice-versa). These connections between these logical notions are important. Although validity1 (and equivalence1 ) are more significant notions in the overall architecture of the theory, we can establish them by way of establishing validity2 (and equivalence2 ). 7 Consequences for the logic of will Having acquired a notion of validity, we are ready to explore the main consequences of our apparatus. We start by vindicating our second desideratum, i.e. the scopelessness of will. First, we notice that the semantics satisfies the excluded middle property: Will Excluded Middle. ðwill f A ∨ will f not Añ is valid2 (hence valid1 ). We use ‘∨’ for disjunction to highlight that the sentence is valid on the assumption that or is boolean disjunction.19 The argument for excluded middle also establishes a related fact: Negation Swap. ðwill f not Añ and ðnot will f Añ are equivalent2 (hence equivalent1 ) This explains why we do not perceive different scopes for negation despite the fact that will is a modal. The analysis also entails that will-conditionals satisfy a principle of conditional excluded middle. Interestingly, this holds whether or not conditionals in general satisfy this principle. Compositional CEM for will-Conditionals. For any point 〈w,s, g〉, ¹(if B)(will f A)ºw,s,g = 1 or ¹(if B)(will f not A)ºw,s,g = 1 Note that this and all the following about conditionals are restricted to the case in which the conditional antecedent is compatible with the modal base. Since Compositional CEM holds at any point of evaluation,20 we get: 19 PROOF: let 〈w,s, g〉 be an arbitrary point of evaluation. We have that ðwill f A ∨ will f not Añ is true at 〈w,s, g〉 iff either A is true at s(w,F) or false at s(w,F). But the right-hand side of the biconditional is always true (since, for any set of worlds S and any world w, it is always the case that w either belongs or doesn’t belong to it). Hence, ðwill f A ∨ will f not Añ is true at 〈w,s, g〉, hence ðwill f A ∨ will f not Añ is true at any point of evaluation. 20 Whether ð(if B)(will f A)ñ is true at 〈w,s, g〉 boils down to the truth of A at 〈w,s, g[ f 7→ F ∩ B]〉. If it is not true, it must be false, but in that case, ð(if B)(will f not A)ñ must be true. 21 Postsemantic CEM for will-Conditionals. ð(if B)(will f A) ∨ (if B)(will f not A)ñ is valid2 (hence valid1 ) Relatedly, there is only one way of negating the consequents of conditionals. Narrow Negation Swap in Conditionals. ð(if B)(will f not A)ñ and ð(if B)(not will f A)ñ are equivalent2 (and hence equivalent1 ). This is a trivial consequence of the non-conditional negation swap, because (if B) merely operates on g. More importantly, we can derive: Wide Negation Swap in Conditionals. ð(if B)(will f not A)ñ and ðnot (if B)(will f A)ñ are equivalent2 (and hence equivalent1 ).21 This completes our illustration of the basic logical implications of our semantics as well as our explanation of how modal theorists can explain the scopelesness of will. 8 8.1 Belief and doubt in will-claims Probabilities of simple will-claims Recall the cognitive problem from §2.3. Ordinary agents are uncertain about the future. On one natural way to understand this uncertainty, this means that ordinary agents have nonextreme degrees of belief in the propositions expressed by will-claims. Moreover, at least in some cases, it seems that this uncertainty is rationally permissible, if not rationally required. An adequate theory of will should vindicate this intuition. This problem should be distinguished from a different, important problem surrounding belief in future claims. Suppose that the future is objectively open, in the sense defined above. In particular, suppose that branching theorists in the style of Thomason (1970), Belnap & Green (1994), Belnap et al. (2001) are right about the metaphysics of branching: there are several possible futures, each of which share the segment that we occupy at the present time. In this case, it is unclear what we mean when we say that the probability of an open proposition— say, the proposition that Cynthia will wear a Warriors cap tomorrow—is r. If it is genuinely open whether Cynthia will wear a Warriors cap, then there are (at least) two ‘equally real’ futures. In one of them she wears the cap, while 21 PROOF: ¹not (if B)(will f A)ºw,s,g = 1 iff ¹(if B)(will f A)ºw,s,g = 0 iff ¹will f Aºw,s,g[ f 7→F∩B] =0 iff ¹not will f Aºw,s,g[ f 7→F∩B] =1 iff ¹(if B)(not will f A)ºw,s,g =1. Together with Narrow Negation Swap, this equivalence entails Wide Negation Swap. 22 Will do better in the other she does not. This problem has received attention in philosophy of physics (in particular, by defenders of the Everett interpretation of quantum mechanics—see, among others, Wallace 2014). We are not contributing to this project here. Rather, we are taking for granted that it somehow makes sense to assign nonextreme credences to propositions about the future (and hence, derivatively, to the claims that express them).22 Let us then return to our Sports Fan example. Recall: every day, Cynthia tosses a fair die and, on the basis of the outcome, decides whether to wear a Giants hat, a Warriors hat, or no hat. Consider a rational agent who assigns credence 1/3 to each of the three possibilities. Against this background, what we want to show is that the proposition expressed by (10) (repeated below) on our account also gets credence 1/3. (10) Cynthia will wear a Warriors cap tomorrow Before we start, let us remind you that this is a nontrivial task. In fact, as we pointed out in §2.4, contemporary accounts in the linguistic semantics tradition fail this task. The reason is that these accounts declare ðIt will be the case that Añ true just in case A is true at every possible future. This semantics makes (10) false, and hence predicts that a rational agent should assign credence zero to the proposition it expresses. Let us spell out some basic assumptions. In keeping with standard possible worlds semantics for attitudes, assume that credences are defined over sets of worlds. In particular, assume that an agent’s credences at a given time may be modeled by a probability function µ satisfying the usual constraints. For example, let µ model an agent’s credences at the current point in time. Let µ(A) = 1/3, where A is the set of worlds where Cynthia wears a Warriors cap. Our task is to check that µ(PROPW ) = 1/3, where PROPW is the content our semantics associates to an utterance of (10). For current purposes, we can take the content expressed by the utterance of a will-sentence at a given context to be the set of worlds such that the utterance is true as evaluated at those worlds. Formally: Content of A at c: kAkc = {w : ¹Aºw,sc ,g c = 1} In what follows, we suppress reference to the context to avoid clutter. 22 If you’re skeptical about these claims, you may take our arguments in this section as providing a good litmus test for our semantics of will. What we’re going to show is that if our theories of credences warrants assigning the attitudes that seem intuitive offhand, then the semantics delivers contents that vindicate the intuitive assignment of credences. 23 it is easy to see that this yields exactly the verdict we need. Take our example: kCynthia will wear a Warriors capk is just the set of worlds in which Cynthia wears a Warriors cap. On the assumption that the credence that our agent assigns to Warriors-cap-worlds is 1/3, she will also assign credence 1/3 to the proposition expressed by (10). More generally, we obtain the result that, for any prejacent A, kwill f Ak = kAk. 8.2 Probabilities of complex will-sentences The probabilities of Boolean compounds of will-sentences work out as one would intuitively expect. In virtue of our earlier result that the scope of negation does not matter, we have that: knot will f Ak = {w : w is an ¬A-world} = k¬Ak In our example, kIt is not the case that Cynthia will wear a Giants capk = kCynthia will not wear a Giants capk. Both propositions have probability 2/3 (according to µ). Similar results holds for conjunction and disjunction, so that: kwill (A & B)k = kwill Ak ∩ kwill Bk and kwill (A ∨ B)k = kwill Ak ∪ kwill Bk. The case of conditionals is more complex. It is well-known that standard possible worlds semantics for conditionals fails to vindicate, in general, the intuitive assignments of probabilities to conditional sentences.23 Since our account of will mirrors one of the standard semantic proposals for conditionals, it shares this feature. It is helpful to give a sense of how standard semantics for conditionals fails to vindicate intuitive probability judgments. Consider again the Sports Fan scenario, and take the conditional: (24) If Cynthia wears a cap, she will wear a Warriors cap. Recall that Cynthia decides to wear a Warriors cap, a Giants cap, or no cap, depending on a random process that makes each of the three options 1/3 likely. Accordingly, suppose that (24) is evaluated against the toy modal base we described in section 3, and consisting of a Warriors-cap-world, a Giants-cap-world, and a no-cap-world. It seems natural to say that your degree of belief in (24) should be (or, at the very least, may be) 1/2. But our semantics can’t vindicate this result. 23 See Lewis (1979b), Lewis (1986b), Hájek & Hall (1994). It is often assumed that the intuitive probabilities of conditional sentences should match the conditional probabilities of the consequent, given the antecedent. But one doesn’t need to endorse this general thesis (which has been called into question; see e.g. Kaufmann (2004)) to see the problem. All we need is that there are examples in which such an assignment is plausible, such as (24) in the main text. 24 Will do better To see this, consider the content of (24): (25) kIf cap, will f Warriors capk = {w: ¹If cap, will f Warriors capºw,s,g = 1} = {w: ¹will f Warriors capºw,s,g[ f →F∩cap] = 1} = {w: ¹Warriors capºs(w,F∩cap),s,g[ f →F∩cap] = 1} A world belongs to this proposition if the selection function maps to it when given as input that world and the restricted modal base. it is easy to show that the proposition in (25) cannot have probability 1/2. The basic point is that, given that in our model we have only three worlds, each of which has probability 1/3, no proposition (i.e, no set of worlds) can have probability 1/2.24 One could respond that the problem depends on the fact that we have used a modal base that is too simple. If we add enough worlds to the modal base, we will be able to assign to (25) a content that gets probability 1/2. This is correct, but the point illustrated by our toy example will still hold for some conditional or other. Alan Hájek (1989, 2012) has pointed out that for any (nontrivial and finite-ranged) probability function P r, the conditional probability values assigned on the basis of P r outnumber the unconditional probability values assigned by P r. Hence there will always be some conditional probability value that doesn’t find a match in the probability of any proposition—exactly as it happens in our example, where no proposition has probability 1/2. Before closing, let us gesture towards a way of refining our ideas that will yield better results for conditionals. We build on a line of thinking of conditionals that has been developed over the past three decades (McGee 1989, Jeffrey & Stalnaker 1994, Bradley 2012). So far, we have assumed that the credences of a rational agent are distributed over an algebra of possible world propositions. But there is a second dimension of uncertainty, which this model doesn’t capture: an agent may be uncertain about which world is selected by the selection function. Go back to our example and consider the no-cap-world z. What is the value of the selection function, when the relevant arguments are z and the proposition that Cynthia wears a cap? Both the answers ‘the Warriors-cap-world w’ and ‘the Giants-cap-world v’ seem to be open epistemic possibilities. Being uncertain between these two answers, of course, would mean being uncertain about which 24 PROOF: The modal base contains three worlds: the Warriors-cap-world w, the Giants-cap-world v, and the no-cap-world z; each has probability 1/3. w is a member of k(24)k and v is not. The question is whether z is. This depends on the value of the selection function when z is the input world. If s(z,F ∩ cap) is w, then z is also a member of k(24)k; otherwise, not. In the former case, the probability of (24) is 2/3, and in the latter 1/3. Either way, that probability is different from 1/2. 25 between two candidates for the selection function (call them ‘sw ’ and ‘s v ’) is the ‘correct’ one. To register this kind of uncertainty alongside uncertainty about which world is actual, we need to refine the elements of the underlying algebra. Rather than worlds, we may use pairs of consisting of a world and a selection function. This also involves modifying our notion of content: we need to take contents to be sets of pairs of a world and a selection function.25 0 0 2D content of A at c: 〈〈A〉〉 = {〈w0 ,s0 〉 : ¹Aºw ,s ,g c = 1} It is straightforward to see how, on the new picture, (24) may be assigned probability 1/2. We now have an algebra of six possibilities, consisting of the pairs: {〈w,sw 〉,〈v,sw 〉,〈z,sw 〉,〈w,s v 〉,〈v,s v 〉,〈z,s v 〉} it is easy to show that probability distribution that assigns to each of these pairs probability 1/6 assigns probability 1/2 to (24). 9 Conclusion Traditionally, will has been treated as a tense in philosophy, and as a modal in linguistics. Linguists are right: there is conclusive evidence that will is a modal. At the same time, all existing modal theories fail to deliver some important desiderata. In particular, they cannot be integrated with our intuitive attitudes towards the future. We have suggested that will is indeed a modal, but doesn’t have a quantificational semantics. Rather, will selects the ‘one actual future’ out of the set of historical alternatives at the time of utterance. Besides validating the evidence for the modal character of will, this account predicts a range of important logical interactions for will and dovetails well with intuitions about the cognitive role of future statements—thus doing better than any other theory on the market. 25 See Bradley (2012) for a much more extensive development of the idea. 26 Will do better References Abusch, Dorit. 1997. Sequence of tense and temporal de re. Linguistics and Philosophy 20(1). 1–50. Abusch, Dorit. 1998. Generalizing tense semantics for future contexts. In Rothstein S. (ed.), Events and Grammar (volume 70 of Studies in Linguistics and Philosophy, 13–33. Kluwer. Barnes, Elizabeth & Ross Cameron. 2009. The open future: bivalence, determinism and ontology. Philosophical Studies 146. 291–309. 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